#10407800

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#53702	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4035 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,9)(4,8)(5,6)(7,10)(11,16)(12,14)(13,15)(33,34)(35,36)(37,45)(38,40)(39,47)(41,43)(42,48)(44,46);(2,6)(3,10)(8,14)(11,13)(17,18,22,20)(19,25,27,23)(21,28,29,24)(26,30,32,31)(33,35,38,43)(34,36,40,41)(37,46,47,42)(39,48,45,44);(4,12)(8,14)(11,13)(15,16)(18,20)(23,25)(24,28)(30,31)(33,34,38,40)(35,41,43,36)(37,39,47,45)(42,48,46,44);(1,3)(2,7)(4,13)(5,10)(6,9)(8,16)(11,12)(14,15)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,46)(47,48);(1,4)(2,8)(3,11)(5,12)(6,14)(7,15)(9,16)(10,13)(17,20,22,18)(19,23,27,25)(21,28,29,24)(26,30,32,31)(33,34)(35,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,37)(34,39)(35,42)(36,44)(38,47)(40,45)(41,48)(43,46)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=hi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10314996);